# std::beta, std::betaf, std::betal

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float beta ( float x, float y ); double beta ( double x, double y ); long double beta ( long double x, long double y ); |  | (since C++17) (until C++23)
/* floating-point-type */ beta( /* floating-point-type */ x, /* floating-point-type */ y ); |  | (since C++23)
float betaf( float x, float y ); | (2) | (since C++17)
long double betal( long double x, long double y ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Arithmetic1, class Arithmetic2 > /* common-floating-point-type */ beta( Arithmetic1 x, Arithmetic2 y ); | (A) | (since C++17)

1-3) Computes the Beta function of x and y. The library provides overloads of `std::beta` for all cv-unqualified floating-point types as the type of the parameters x and y.(since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

### Parameters

x, y | - | floating-point or integer values

### Return value

If no errors occur, value of the beta function of x and y, that is \(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)∫1
0tx-1
(1-t)(y-1)
dt, or, equivalently, \(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)

Γ(x)Γ(y)
Γ(x+y)

is returned.

### Error handling

Errors may be reported as specified in math_errhandling.

- If any argument is NaN, NaN is returned and domain error is not reported.
- The function is only required to be defined where both x and y are greater than zero, and is allowed to report a domain error otherwise.

### Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of this function is also available in boost.math.

std::beta(x, y) equals std::beta(y, x).

When x and y are positive integers, std::beta(x, y) equals \(\frac{(x-1)!(y-1)!}{(x+y-1)!}\)

(x-1)!(y-1)!
(x+y-1)!

. Binomial coefficients can be expressed in terms of the beta function: \(\binom{n}{k} = \frac{1}{(n+1)B(n-k+1,k+1)}\)⎛
⎜
⎝n
k⎞
⎟
⎠=

1
(n+1)Β(n-k+1,k+1)

.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1 and second argument num2:

If num1 or num2 has type long double, then std::beta(num1, num2) has the same effect as std::beta(static_cast<long double>(num1), static_cast<long double>(num2)). Otherwise, if num1 and/or num2 has type double or an integer type, then std::beta(num1, num2) has the same effect as std::beta(static_cast<double>(num1), static_cast<double>(num2)). Otherwise, if num1 or num2 has type float, then std::beta(num1, num2) has the same effect as std::beta(static_cast<float>(num1), static_cast<float>(num2)). | (until C++23)
If num1 and num2 have arithmetic types, then std::beta(num1, num2) has the same effect as std::beta(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, arguments of integer type are considered to have the same floating-point conversion rank as double. If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided. | (since C++23)

### Example

Run this code

    #include <cassert>
    #include <cmath>
    #include <iomanip>
    #include <iostream>
    #include <numbers>
    #include <string>

    long binom_via_beta(int n, int k)
    {
        return std::lround(1 / ((n + 1) * std::beta(n - k + 1, k + 1)));
    }

    long binom_via_gamma(int n, int k)
    {
        return std::lround(std::tgamma(n + 1) /
                          (std::tgamma(n - k + 1) *
                           std::tgamma(k + 1)));
    }

    int main()
    {
        std::cout << "Pascal's triangle:\n";
        for (int n = 1; n < 10; ++n)
        {
            std::cout << std::string(20 - n * 2, ' ');
            for (int k = 1; k < n; ++k)
            {
                std::cout << std::setw(3) << binom_via_beta(n, k) << ' ';
                assert(binom_via_beta(n, k) == binom_via_gamma(n, k));
            }
            std::cout << '\n';
        }

        // A spot-check
        const long double p = 0.123; // a random value in [0, 1]
        const long double q = 1 - p;
        const long double π = std::numbers::pi_v<long double>;
        std::cout << "\n\n" << std::setprecision(19)
                  << "β(p,1-p)   = " << std::beta(p, q) << '\n'
                  << "π/sin(π*p) = " << π / std::sin(π * p) << '\n';
    }

Output:

    Pascal's triangle:

                      2
                    3   3
                  4   6   4
                5  10  10   5
              6  15  20  15   6
            7  21  35  35  21   7
          8  28  56  70  56  28   8
        9  36  84 126 126  84  36   9

    β(p,1-p)   = 8.335989149587307836
    π/sin(π*p) = 8.335989149587307834

### See also

tgammatgammaftgammal (C++11)(C++11)(C++11) | gamma function (function)

### External links

Weisstein, Eric W. "Beta Function." From MathWorld — A Wolfram Web Resource.

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